Robust Control Toolbox™

Grid ureal parameters uniformly over their range

Syntax

• B = gridreal(A,N) 
  [B,SampleValues] = gridreal(A,N) 
  [B,SampleValues] = gridreal(A,Names,N) 
  [B,SampleValues] = gridreal(A,Names1,N1,Names2,N2,...) 
  

Description

B = gridureal(A,N) substitutes N uniformly-spaced samples of the uncertain real parameters in A. The samples are chosen to cut "diagonally" across the cube of real parameter uncertainty space. The array B has size equal to [size(A) N]. For example, suppose A has 3 uncertain real parameters, say X, Y and Z. Let (x1, x2 , , and xN) denote N uniform samples of X across its range. Similar for Y and Z. Then sample A at the points (x1, y1, z1), (x2, y2, z2), and (xN, yN, zN) to obtain the result B.

If A depends on additional uncertain objects, then B will be an uncertain object.

[B,SampleValues] = gridureal(A,N) additionally returns the specific sampled values (as a structure whose fieldnames are the names of A's uncertain elements) of the uncertain reals. Hence, B is the same as usubs(A,SampleValues).

[B,SampleValues] = gridureal(A,NAMES,N) samples only the uncertain reals listed in the NAMES variable (cell, or char array). Any entries of NAMES that are not elements of A are simply ignored. Note that gridureal(A, fieldnames(A.Uncertainty),N) is the same as gridureal(A,N).

[B,SampleValues] = gridureal(A,NAMES1,N1,NAMES2,N2,...) takes N1 samples of the uncertain real parameters listed in NAMES1, and N2 samples of the uncertain real parameters listed in NAMES2 and so on. size(B) will equal [size(A) N1 N2 ...].

Example

Create two uncertain real parameters gamma and tau. The nominal value of gamma is 4 and its range is 3 to 5. The nominal value of tau is 0.5 and its value can change by +/- 30 percent.

• gamma = ureal('gamma',4); 
  tau = ureal('tau',.5,'Percentage',30); 
  

These uncertain parameters are used to construct an uncertain transfer function p. An integral controller, c, is synthesized for the plant p based on the nominal values of gamma and tau. The uncertain closed-loop system clp is formed.

• p = tf(gamma,[tau 1]); 
  KI = 1/(2*tau.Nominal*gamma.Nominal); 
  c = tf(KI,[1 0]); 
  clp = feedback(p*c,1); 
  

The figure below shows the open-loop unit step response (top plot) and closed-loop response (bottom plot) for a grid of 20 values of gamma and tau.

• subplot(2,1,1); step(gridureal(p,20),6) 
  title('Open-loop plant step responses') 
  subplot(2,1,2); step(gridureal(clp,20),6)
  
  
  

It clearly illustrates the low-frequency closed-loop insensitivity achieved by the PI control system.

Multi-Parameter Example

The next example illustrates the different options in gridding high-dimensional (e.g., n greater than 2) parameter spaces. An uncertain matrix, m, is constructed from four uncertain real parameters, a, b, c and d, each making up the individual entries.

• a=ureal('a',1); b=ureal('b',2); c=ureal('c',3); d=ureal('d',4); 
  m = [a b;c d]; 
  

In the first case, the (a,b) space is gridded at five places, and the (c,d) space at 3 places. The uncertain matrix m is evaluated at these 15 grid-points resulting in the matrix m1.

• m1 = gridureal(m,{'a';'b'},5,{'c';'d'},3); 
  

In the second case, the (a,b,c,d) space is gridded at 15 places, and the uncertain matrix m is sampled at these 15 points. The resulting matrix is m2.

• m2 = gridureal(m,{'a';'b';'c';'d'},15); 
  

The (2,1) entry of m is just the uncertain real parameter c. Below, you see the histogram plots of the (2,1) entry of both m1 and m2. The (2,1) entry of m1 only takes on 3 distinct values, while the (2,1) entry of m2 (which is also c) takes on 15 distinct values uniformly through its range.

• subplot(2,1,1) 
  hist(m1(2,1,:)) 
  title('2,1 entry of m1') 
  subplot(2,1,2) 
  hist(m2(2,1,:)) 
  title('2,1 entry of m2')
  
  
  
  

See Also
usample     Generates random samples of an atom

usubs       Substitutes values for atoms

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